Yes it does because there are no boundaries. If there are boundaries, then it's not infinite.

If you claim there are infinite apples and I find a place with no apples, then you don't have sufficient apples to go around in order to fill every place and therefore the quantity of apples is limited or else there would be apples in every place that apples could exist.

If there could be an apple here, but there is no apple here, then the only reason to explain that is insufficient apples.

If I would claim there are infinite apples available, then yes that would apply. Also with one apple. But to have infinite apples in an infinite universe would not necessarily guarantee any apples within reach.

Like if time is infinite, which it is since beginnings are part of the concept time, then even if you are born now, some of your actions will still have effects for an infinite duration, unless the universe collapses into a singularity which I think is fantasy.

So your deeds would (do) resound in the infinite future, but not in the also infinite past.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

The whole thing is boringly easily solved when you see that infinite is a predicate and not a proposition.

So "infinity" of itself just means either everything or nothing. But any meaningful statement with infinity in it still gives infinity as a limited part of the proposition, where quality x or y or z is required to make the proposition.

"Does infinity exist" must mean "is existence infinite?"

And if it is, then all things must somehow reach into infinity too. And they do, by their consequences in time.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

So the Arabs who came up with algebra (with the contributions of a guy called Al Jabr) and also with 0 destroyed the Euclidean and Parmenidean idea of numbers as elements of a world, and made them magical appearances defying apparent reality yet working very well with the brain.

But this defiance can be because whats defied is what wasn't proven yet. Maybe they were like hold on, its taken pretty long now that we sought for the end-all, and we didn't find it so lets just suppose it goes on forever. Then at least we can be free of this supposed end-all and appreciate what we really see.

In my understanding, both 0 and infinite are predicates, and for X=existence they give respectively false and true because otherwise the concepts are contradicted.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

If the axiom is "something exists" then for 0 to be true there needs to be a thing that is withdrawn from another. For infinity to be true there needs only be one thing, it just needs to be infinite. So 0 is a higher order operator than infinity.

Still, infinity is a higher order principle to 1.

2 and uncountability are the same by implication because to get from 1 to 2 you need an assumption, which is that things are separate and not unified, and if you do that there is certainly no way of demonstrating any limits to the number of things that can be listed. And infinity is a higher order function of uncountability.

Only after all this is fixed we get to 0.Or while its not being fixed but then 0 is seen as the basic depth of the thing where it is actually the summit.

Infinity is the root of all hypothetical numbers, including 0. 1 is the only non hypothetical number because it is the only number which can contain all others.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

barbarianhorde wrote:If the axiom is "something exists" then for 0 to be true there needs to be a thing that is withdrawn from another.

????. Doesn't parse at my end.

barbarianhorde wrote:For infinity to be true

Infinity can't be true or false. It's not a thing that can be true or false. A set being infinite might be true or false. Precision is critical.

barbarianhorde wrote:there needs only be one thing, it just needs to be infinite. So 0 is a higher order operator than infinity.

Doesn't make sense to me.

barbarianhorde wrote:Still, infinity is a higher order principle to 1.

Order of what? Haven't seen order defined.

barbarianhorde wrote:2 and uncountability are the same by implication

Nonsense.

barbarianhorde wrote:because to get from 1 to 2 you need an assumption, which is that things are separate and not unified, and if you do that there is certainly no way of demonstrating any limits to the number of things that can be listed.

More nonsense. I have one apple, I have two apples. I have no idea what you are talking about. Have you a reference so I can have some clue as to what domain of discourse you're working in?

barbarianhorde wrote:And infinity is a higher order function of uncountability.

Doesn't parse. Says nothing. Word salad.

barbarianhorde wrote:Only after all this is fixed we get to 0.

After all what is fixed? What's broken?

barbarianhorde wrote:Or while its not being fixed but then 0 is seen as the basic depth of the thing where it is actually the summit.

You often make sense. This post of yours does not make any sense.

barbarianhorde wrote:Infinity is the root of all hypothetical numbers,

What is a hypothetical number? What other kinds of numbers do you have in mind that aren't hypothetical? The root? Like the root of a polynomial, or a square or cube root? You're just throwing out random words. This is unlike your usual posts, which are generally connected with reality and sense.

barbarianhorde wrote: including 0. 1 is the only non hypothetical number because it is the only number which can contain all others.

barbarianhorde wrote:So "infinity" of itself just means either everything or nothing.

Isn't the collection of even numbers infinite? They're clearly only a part of something larger. They're not everything. They don't include the odd numbers, for example.

I see you wrote several posts, not just one. But you seem to have decided to wake up this morning and post strings of word salad, devoid of meaning or sense. I don't mean for that to be an attack. Only an observation. I've come to expect sensible posts from you. If you only posted nonsense I wouldn't bother to mention it.

No. Consider a variable x that ranges over the set {1, 2, 3}. Consider basic finite probability theory. The roll of a single six-sided die. You use variables to stand for things like "I roll a 3," or "I roll and even number." There is no implication of infinity.

barbarianhorde wrote:In my understanding, both 0 and infinite are predicates,

No. What can you possibly mean by that?

barbarianhorde wrote: and for X=existence they give respectively false and true because otherwise the concepts are contradicted.

Word salad. Makes no sense. I'm disturbed by the fact that I formerly thought you were making some level of sense in your posts, and now I wonder if I missed this strain of illogic. Can you put your morning flood of posts into context? It all seems ... well, not good.

lol, yeah this is why you're not a philosopher. I managed to keep it extremely simple for you, go along in your little baby steps, doing a bit of theatrics, that was when you thought I was making sense.

I should take offence at your radical laziness but I know mathematicians hold this for some sort of virtue. Ive been ahead of you constantly, drawn your proud definitions and drawings out of you by pretending I didn't understand so well, telling you the difficulties along the way. This is all because I don't think inside of language but just use language where it is constructive.

Now Ill leave you to your graceful temper.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

Serendipper wrote:Infinity has a habit of eternally popping up in debates, so I figured I'd put together a thread that is easily referenced upon such occurrence that will dissuade folks from religiously promulgating the concept of infinity as an explanation for the unexplainable.

First, what is it?

infinite[in-fuh-nit]

adjective1. immeasurably great.2. indefinitely or exceedingly great.3. unlimited or unmeasurable in extent of space, duration of time, etc.4. unbounded or unlimited; boundless; endless.5. Mathematics: not finite. (of a set) having elements that can be put into one-to-one correspondence with a subset that is not the given set.

According to definition #1, there is a sense in which the infinite can describe merely what is not measurable, so in that light, finite amounts can be so large that they are not measurable, yet are still finite. This is not the sense that I intend to deal with when talking about infinity. Finite numbers that are so large that they couldn't be represented on the entirety of the observable universe, even if written on the planck scale, I'm defining those as "dark numbers" because they are finite, but unrepresentable (dark/unseen) within the universe.

Hopefully we can all agree that a good working definition of infinity is "boundless", "without bounds or constraints: either physical or conceptual".

Now, can the boundless exist? Well, what does it mean to exist? This dot exists ---> . because there is something that is not-dot providing contrast and context (the white background), so existence is the relationship of the dot to the not-dot because if either are missing, then there is no dot and no dot could be said to exist. Existence is therefore dependent upon relationship and relationship precedes concepts of existence.

Now, what if the dot had no boundary? Well, immediately we can surmise that it would have no contrast because if the dot had no boundary, there would be nothing that is not-dot to provide the contrast in order to underpin existence. And if there were something to provide contrast, then obviously the dot would have a boundary. So right off the bat we can say infinity isn't anything that can exist, but I'm just getting started.

Infinities are said to contain things, because they contain infinite things, but how can a container contain anything with no walls (boundaries)?

Infinities cannot have beginnings or ends because those are boundaries and we said in the beginning that infinity has no boundaries. We cannot divide infinity in half and say infinity is bounded by this finite location and extends to infinity in that direction; it's nonsense and breaks our definition of infinity being boundless. Zero is not a boundary, but is just an arbitrary starting point on an infinite number line extending in both directions and we could just as easily started at -2,-1,0,1,2,3,etc or 5,6,7,8,etc. A line that is not infinite is a segment because all lines are defined to be infinite within the construct of mathematics; therefore a line with a beginning (such as a timeline) is not an example of infinity. Further, if time had a beginning, infinite time could not be said to exist until forever arrived, and forever means never because forever can never be realized, so infinite time could never exist if time had a beginning.

A better conceptualization for eternity is absence of time instead of infinite amounts of it, but really they both mean the same thing since in both cases time would have no relevance.

I disagree here. Eternity can be taken to be implicit in the concept of time, given that this concept includes all beginnings and ends. "Time began" is a problematic idea. Eternity is an infinity of moments, the idea of an unfolding dimension made subjective, tied to a reference frame.

Absent reference frames there is not eternity but chaos.

Since infinity has no starting point/reference point/unique edge, then infinite computer memory would equate to having no memory and anything written to memory could never be found again. Where would allocation start? Afterall, the memory stick would take every bit of space in the entire universe because to say it wouldn't would be to limit the size of it. Where would an origin/center be placed and how could it be found again?

The infinite is the ubiquitous, omnipresence.

Ah! This is where I really disagree.

First of all, "The" infinite is not the same as Infinity.

Infinity is a predicate to a given, it is not a starting point. If you presume it and ask if it exists, thats the wrong order.

You must observe things and then ask if they are infinite to ground it in reason.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

The conclusion all along was very old, that if you're committed to abstracting empirical data all the way, unlike the Greeks were, the ideas of infinity and 0 become available.

Sets are just ultra lazy abstractions without any class or style. Russell at least had class and style, which is why he pricked through set theory in one gesture, and liked Wittgenstein, who is a fledgeling philosopher in how he overcame his Tractatus.

Serendipper for the win because he exposed the grammatical naiveté which causes all the various perspectives unawareness of being various.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

Whats cool about being all too realistic before infinity is that you're probably very aware of the finitude of some pretty important shit. Its possible to arrive at the value of finite things, not of infinite things.and since there is definitely value that can be attributed to the power to identify a definite value, I think you can't do things like quantum computing on set theory.

Anyway, this is very interesting. Set theory fails. Type theory must take its place. Thats a lot of grinding fucking weight lifting.

Blimey if I cant see now why mathematicians always lie on couches so proudly and don't walk around to think. Set theory.

Ah, lets say we have a set, blah, and a set blah, and oh what would they do together! Oh what a delightful rainbow of things!

No, not things. Dreams.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

barbarianhorde wrote:If I would claim there are infinite apples available, then yes that would apply.

X apples available would be theoretical and not actual. You either have in possession/existence the number of apples or you don't.

But to have infinite apples in an infinite universe would not necessarily guarantee any apples within reach.

Why not? Do you mean to say there is a possibility that extra space could exist such that there would be insufficient apples to fill it? In that case there would not be infinite apples. Conversely, if there are infinite apples, then there cannot be extra space left over.

To say one infinity is bigger than another is to place limits on the smaller infinity which would then make it finite.

Like if time is infinite, which it is since beginnings are part of the concept time,

If time is infinite, then having a beginning is impossible. This is the same as my argument against having an infinite road that has a beginning.

barbarianhorde wrote:Of course it would be easy to negate infinity if you start out with the assumption of a finite universe.

You can just say "the universe is finite, thus it isn't infinite, thus nothing in it is infinite".But who will think this makes sense? Only people who already agreed with you on faith.

You can't prove an end to the universe, or to a straight line, so to insist it does have one (i.e isn't infinite) is like being really ambitious without any means.

The universe must be finite or else it wouldn't be definite and therefore wouldn't exist. Plus, there is no evidence to indicate the universe is infinite and lots of evidence to preclude it, such as: the conservation of energy which wouldn't make any sense in infinite energy, the fact that a photon cannot be emitted until its partner is found across spacetime which wouldn't be possible if parts of spacetime were infinitely far away, the fact that the speed of light is a definite (finite) number where time and space end.

So, in summary:

- Lots of evidence suggesting a finite universe- No evidence suggesting an infinite universe- Yet people still believe the universe is infinite

barbarianhorde wrote:I disagree here. Eternity can be taken to be implicit in the concept of time,

You're assuming time means causality, but it is not since causality cannot describe the speed of causality. Time is something that can speed up and slow down, but causality simply describes one thing leading to another with no speed component.

given that this concept includes all beginnings and ends.

There can't be beginnings or ends in infinite anything.

"Time began" is a problematic idea.

Time without a beginning is a problematic proposition.

Eternity is an infinity of moments,

How long is a moment? 1s/infinity? lol

the idea of an unfolding dimension made subjective, tied to a reference frame.

Yes, subjectivity is the universe in reference to another part of the universe.

Absent reference frames there is not eternity but chaos.

Chaos is deterministic, but sensitive to initial conditions. Randomness is the uncaused event.

First of all, "The" infinite is not the same as Infinity.

What's the difference?

Infinity is a predicate to a given, it is not a starting point. If you presume it and ask if it exists, thats the wrong order.

You must observe things and then ask if they are infinite to ground it in reason.

Gloominary wrote:It's not logically or grammatically incorrect to say a wall is infinitely tall, or infinitely wide, a thing can be infinite in some quality without having to be infinite in all qualities.

Whether it's empirically incorrect is another matter.

Infinitely wide is ok because it extends in both directions, but infinitely tall would be a wall that extends around the universe until it connected with the other side of the earth.

Why would the infinitely tall wall curve around the cosmos and touch its bottom?

The same as you, but there can be an absolute infinity, and specific infinities.

Unlimited in some ways and limited in others is unlimited in quantity and limited in identity/category.

Or unlimited in quantity here/now, but limited there/then, or unlimited in x qualities, but not in y.

I can't conceptualize it and you can't either. In actuality, where in the universe do you suspect that it may be possible to draw an infinite line in one direction, but not in the other?

Conceptually you can draw a line anywhere, but you can also conceive of a road ending one way, but not the other.

You can also conceive of an impenetrable wall that keeps everything this side of it from crossing over, not that such a wall is necessary for a road to end one way, but not the other.

Beyond the wall, there might be nothing, not merely empty space, but no MEST at all, or there might be stuff.

Then, at this moment, it is not infinite because there exists a place for more road.

It's endless backwardly, and endful forwardly.

How can you propose having an infinite road when clearly we could make it longer?

You can make it longer forwardly, but not backwardly.

A road that is truly infinite would extend around and around the universe many many times until it occupied every planck cube in the universe, completely displacing all matter, and until it eventually connected with itself for lack of having anywhere else to go. To say that isn't so is to say the road has a boundary which would make it not infinite.

It has a boundary backwardly, but not forwardly.

Yes it does matter because if there is a place for another apple, but no apple is there, then we have found a boundary and therefore the number of apples is not infinite.

Apples could unendingly sparsely populate the unending universe, and still be unending in number, which means some infinites could be bigger than others.

Size must have a zero like temperature and speed or else it couldn't exist. We can't get infinitely colder, infinitely slower, infinitely smaller and if we could, then temperature, speed, size would have no significance/meaning.

I'm not so sure, for example, if two things are both infinitely divisible, but, finitely multipliable, if you will, than one of them could still be bigger, stronger and so on than the other.

But even if things are necessarily finitely small, the smallest unit of matter, motion and space might still be centillions of times smaller than quarks.

It seems weird...asymmetrical to me the universe could be infinitely big, but not also infinitely small, and if a thing could be infinitely big, and not infinitely small, than why couldn't a thing be finitely big, and also infinitely small?

MagsJ wrote:Perhaps the fact that it can be imagined is all that infinity needs to exist ...

Question: Do purple flying elephants exist? I can imagin e them.

The possibility of anything we can imagine existing is endless and infinite

Bonus question: Is Ahab captain of the Pequod?

Second bonus question: If George Washington was the first president of the US, and Ahab was the captain of the Pequod, are those statements equally true? True in the same way? Both Washington and Ahab have an equal claim to existence? Along with the purple flying elephants?

George W was not the first president of America, and Ahab was a fictional character, so the first premise is technically not true.. the second is fictionally true, of things that we read in books..

I hope you can see that you need to greatly qualify your remark about imaginability being sufficient for existence.

MagsJ wrote:The possibility of anything we can imagine existing is endless and infinite

I do?

The possibility of anything we can imagine existing is endless and infinite

I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get that time back, and I may need it for something at some point in time. Wait! What?

Serendipper wrote:Pretend you're talking to a 5-year old kid who doesn't know what infinity means. If you say "we have an axiom of infinity", the kid will look at you stupid.

You and I have discussed the axiom of infinity in this thread. Perhaps I'm misremembering. If we have discussed the axiom of infinity, then your remark is disingenuous. If I'm mistaken and we haven't discussed the axiom of infinity, I'll try to remember that I'm talking to a bunch of 5 year olds. That actually explains a lot.

Serendipper wrote:I can integrate an area over a height to yield a volume without using infinity.

That is a very interesting remark. Of course if you took freshman calculus, you can do that using a rote procedure, say by taking an antiderivative of the kind of elementary functions you see in calculus class. Integrand is \(x^2\) so antiderivative is \(\frac{1}{3} x^3\) kind of thing.

But if you studied the subject more deeply, you would realize that in order to form a logically rigorous definition of an integral, you require modern infinitary set theory. In calculus they don't show you that. Perhaps you remember that when they defined the Riemann integral, they defined lower and upper sums relative to a partition, and then you took the LIMIT over all possible partitions. To formalize that requires the full apparatus of ZF set theory, including the axiom of infinity.

So to me, the fact that you DO believe in Riemann integration (aka freshman calculus integration) tells me that you've seen the rote procedures, but not the underlying theory nor all the weird counterexamples and corner cases that made 19th century mathematicians realize they needed a rigorous theory. Infinitary math is essential to define an integral and do freshman calculus. They just don't tell you about this until you take a more advanced course in real analysis.

No infinitary math, no logical foundation for freshman calculus. No axiom of infinity, no Riemann integral.

ps -- Let me give a concrete example. You mentioned integrating an area over a height. How about if you have a rectangular metal plate with a temperature at each point and you want to integrate the temperature over the area of the rectangle to determine the average temperature. You could integrate the vertical slices then the horizontal ones or vice versa. This is multiple integration as in second year calculus. But how do you know when the order of integration matters and when it doesn't? How do you know whether it makes a difference if you integrate the x's and then the y's, or first the y's and then the x's? This can be a very tricky business, especially with a weird or pathological integrand or temperature function. This is when you have to drill down to the rigorous, set-theoretic definition of the integral to prove theorems on reversing the order of integration. In other words the moment you go beyond the simplest examples you need some theory; and the theory of integration requires infinitary set theory, or my name's not Guido Fubini!